How do I figure out my attack bonuses?

Take your base attack bonus for each class the character has and add the first attack bonus together for each. For example, a 1st level Fighter/2nd level Paladin has a total base attack bonus of +3.

To determine if you have multiple attacks per round, use the following: If a given attack’s base attack bonus is +6, there is an additional attack at +1. Every time a new +1 is added to the base attack bonus (from any source) it is applied to all the attacks. When the second attack reaches +6, a third attack appears at +1, and so forth. This way a Paladin12/Blackguard4 will have a base attack bonus of +16/+11/+6/+1

How many spells per day do I get?

Spells per day do not stack except with prestige classes. If you are a Cleric and gain a level of druid, you have two separate pools of spells per day. Conversely some prestige classes such as Shadow Adept (from Forgotten Realms) or Loremaster (from the DMG) add to your current alotment of spells per day. A Wizard10/Loremaster5 has the same number of spells per day as a Wizard15.

This also applies to the potency of a spell. A Cleric6/Druid4 casts Cleric spells at level6 and Druid spells at level 4, whereas a Wizard6/Loremaster4 casts Arcane spells as if he were a Wizard10.

How do I figure out my saving throws?

Add them together. A Fighter1/Rogue1 gets +2 to Fortitude from Fighter and +2 to Reflex from Rogue. Consequently he gets Fortitude+2, Reflex+2, and Will+0 from his class levels.

So what good is a “favored class?”

If you have multiple classes (not prestige classes, just normal classes) and they are more than 1 level apart from each other, one of them had better be your favored class for your race. Otherwise your DM should deduct 20% of your experience earned. This is to prevent people from taking a huge number of classes for one or two levels just for features such as weapon proficiencies (or the Divine Aura of a Paladin).